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## GMAT - Data Sufficiency - Statistics

### At a certain company, a test was given to a group of men and women

At a certain company, a test was given to a group of men and women seeking for promotions. If the average (artihmetic mean) score for the group was 80, was the average score for the women more than 85?

(1) The average score for the men was less than 75.
(2) The group consisted of more men than women.

let the number of men be m and number of women be w, and average score for men be M and average score for woman be W

so,

Total score (T) = m*M + w*W ... (1)

given average score of the group was 80

T = 80*(w + m)  ... (2)

from 1 and 2

m*M + w*W = 80 (w + m)... (3)

Statement 1:

Average score for men was less than 75.

i.e. M<75

if we make M as subject of equation 3

$M=\frac { 80(w\quad +\quad m)\quad -\quad wW }{ m } \le 75$ ... (4)

we need information on w and m to know about W.

Information insufficient.

Statement 2:

The group consisted of more men than women

i.e. m > w

we need information on M to be able to say something. Insufficient.

Taking both statements together

we can write the expression 4 as

$M=\frac { 80(w\quad +\quad m)\quad -\quad wW }{ m } \le 75\\ 80(\frac { w }{ m } )\quad +\quad 80\quad -\quad W(\frac { w }{ m } )\le 75\\ Simplifying,\quad (w/m)(80\quad +\quad W)\quad \le -5\quad ...\quad (5)\\ from\quad statement\quad 2\quad m>w\\ thus,\quad m/w\quad >\quad 1\\ -5*m/w\quad <\quad -5\\ multiplying\quad eq.\quad 5\quad by\quad m/w\quad (greater\quad than\quad 1)\\$

$\\ (80\quad -\quad W)\quad \le \quad -5*m/w\quad <\quad -5\\ 80\quad -\quad W\quad <\quad -5\\ W\quad >\quad 85\\ \\$

hence both statements together are sufficient.

let the number of men be m and number of women be w, and average score for men be M and average score for woman be W

so,

Total score (T) = m*M + w*W ... (1)

given average score of the group was 80

T = 80*(w + m)  ... (2)

from 1 and 2

m*M + w*W = 80 (w + m)... (3)

Statement 1:

Average score for men was less than 75.

i.e. M<75

if we make M as subject of equation 3

$M=\frac { 80(w\quad +\quad m)\quad -\quad wW }{ m } \le 75$ ... (4)

we need information on w and m to know about W.

Information insufficient.

Statement 2:

The group consisted of more men than women

i.e. m > w

we need information on M to be able to say something. Insufficient.

Taking both statements together

we can write the expression 4 as

$M=\frac { 80(w\quad +\quad m)\quad -\quad wW }{ m } \le 75\\ 80(\frac { w }{ m } )\quad +\quad 80\quad -\quad W(\frac { w }{ m } )\le 75\\ Simplifying,\quad (w/m)(80\quad +\quad W)\quad \le -5\quad ...\quad (5)\\ from\quad statement\quad 2\quad m>w\\ thus,\quad m/w\quad >\quad 1\\ -5*m/w\quad <\quad -5\\ multiplying\quad eq.\quad 5\quad by\quad m/w\quad (greater\quad than\quad 1)\\$

$\\ (80\quad -\quad W)\quad \le \quad -5*m/w\quad <\quad -5\\ 80\quad -\quad W\quad <\quad -5\\ W\quad >\quad 85\\ \\$

hence both statements together are sufficient.